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We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of…
The generative modeling of data on manifolds is an important task, for which diffusion models in flat spaces typically need nontrivial adaptations. This article demonstrates how a technique called `trivialization' can transfer the…
Recent studies on metamorphic petrology as well as microstructural observations suggest the influence of mechanical effects upon chemically active metamorphic minerals. Thus, the understanding of such a coupling is crucial to describe the…
The Eulerian distribution on the involutions of the symmetric group is unimodal, as shown by Guo and Zeng. In this paper we prove that the Eulerian distribution on the involutions of the hyperoctahedral group, when viewed as a colored…
In this paper we introduce a model describing diffusion of species by a suitable regularization of a "forward-backward" parabolic equation. In particular, we prove existence and uniqueness of solutions, as well as continuous dependence on…
Diffusion is a commonly used technique for spreading information from point to point on a graph. The rationale behind diffusion is not clear. And the multi-types Galton-Watson forest is a random model of population growth without space or…
The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction-diffusion processes in periodic porous media is analyzed. Our mathematical model can be used for studying…
Solvent-based techniques usually involve preparing dilute blends of electron-donor and electron-acceptor materials dissolved in a volatile solvent. After some form of coating onto a substrate, the solvent evaporates. An initially…
The internal motions of biomatter immersed in biofluid are investigated. The interactions between the fragments of biomatter and its surrounding biofluid are modeled using field theory. In the model, the biomatter is coupled to the gauge…
This is a brief overview of our work on the theory of group invariant solutions to differential equations. The motivations and applications of this work stem from problems in differential geometry and relativistic field theory. The key…
We introduce a deductive statistical mechanics approach for granular materials which is formally built from few realistic physical assumptions. The main finding is an universal behavior for the distribution of the density fluctuations. Such…
Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various…
Living organisms produce metabolites of many types via their metabolisms. Especially, flavonoids, a kind of secondary metabolites, of plant species are interesting examples. Since plant species are believed to have specific flavonoids with…
A Lie groupoid, called \textit{second-order non-holonomic material Lie groupoid}, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference…
The focus of this paper is on the analysis of the Conjugate Gradient method applied to a non-symmetric system of linear equations, arising from a Fast Fourier Transform-based homogenization method due to (Moulinec and Suquet, 1994).…
We work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of homomorphism densities of finite trees and it is…
For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…
Waves play an essential role in many aspects of plasma science, such as plasma manipulation and diagnostics. Due to the complexity of the governing equations, approximate models are often necessary to describe wave dynamics. In this…
Morphogenesis is a developmental process in which cells organize into shapes and patterns. Complex, multi-factorial models are commonly used to study morphogenesis. It is difficult to understand the relation between the uncertainty in the…
We present a thorough study of the differential geometry of weightings and develop the theory of weightings for vector bundles, Lie groupoids, and Lie algebroids. We begin by extending the work of Loizides and Meinrenken on weighted…